reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th80:
  for uf being FinSequence of F_Real st len uf = 3 holds
  <*uf*>@ = 1.(F_Real,3) * (<*uf*>@)
  proof
    let uf be FinSequence of F_Real;
    assume
A1: len uf = 3;
    then
A2: <*uf*>@ = <* <* uf.1 *>, <* uf.2 *> , <* uf.3 *> *> by Th63; then
A3: len (<*uf*>@) = 3 by FINSEQ_1:45;
    set M = 1.(F_Real,3);
    uf is 3-element by A1,CARD_1:def 7; then
A4: uf in REAL 3 by EUCLID_9:2;
    now
A5:   M * <*uf*>@ is Matrix of 3,1,F_Real by A4,EUCLID_8:50,Th74;
      hence len (1.(F_Real,3) * (<*uf*>@)) = 3 by MATRIX_0:23;
      thus (1.(F_Real,3) * (<*uf*>@)).1 = <* uf.1 *>
      proof
        1 in Seg 3 by FINSEQ_1:1; then
A6:     (M * (<*uf*>@)).1 = Line(M * (<*uf*>@),1) by A5,MATRIX_0:52;
        now
          thus len Line(M * (<*uf*>@),1) = width (M * (<*uf*>@))
            by MATRIX_0:def 7
                                        .= 1 by A5,MATRIX_0:23;
          thus Line(M * (<*uf*>@),1).1 = uf.1
          proof
A7:         [1,1] in Indices(M * (<*uf*>@)) by A5,MATRIX_0:23,Th2;
A8:         width M = len (<*uf*>@) by A3,MATRIX_0:23;
            reconsider a1 = 1,a2 = 0 as Element of F_Real;
A9:         Line(M,1) = <* a1,a2,a2 *> & uf =<* uf.1,uf.2,uf.3 *>
              by Th56,A1,FINSEQ_1:45;
            dom uf = Seg 3 by A1,FINSEQ_1:def 3;
            then reconsider uf1 = uf.1, uf2 = uf.2,
              uf3 = uf.3 as Element of F_Real by FINSEQ_1:1,FINSEQ_2:11;
A10:        Line(M,1) "*" uf = a1 * uf1 + a2 * uf2 + a2 * uf3 by A9,Th6
                            .= uf.1;
            1 in Seg 1 by FINSEQ_1:1;
            then 1 in Seg width (M * (<*uf*>@)) by A5,MATRIX_0:23;
            then Line(M * (<*uf*>@),1).1 = (M * (<*uf*>@))*(1,1)
                                            by MATRIX_0:def 7
                                        .= Line(M,1) "*" Col( <*uf*>@,1)
                                            by A7,A8,MATRIX_3:def 4
                                        .= Line(M,1) "*" uf by Th76;
            hence thesis by A10;
          end;
        end;
        hence thesis by A6,FINSEQ_1:40;
      end;
      thus (1.(F_Real,3) * (<*uf*>@)).2 = <* uf.2 *>
      proof
        2 in Seg 3 by FINSEQ_1:1; then
A11:    (M * (<*uf*>@)).2 = Line(M * (<*uf*>@),2) by A5,MATRIX_0:52;
        now
          thus len Line(M * (<*uf*>@ ),2) = width (M * (<*uf*>@))
                                              by MATRIX_0:def 7
                                         .= 1 by A5,MATRIX_0:23;
          thus Line(M * (<*uf*>@),2).1 = uf.2
          proof
A12:        [2,1] in Indices(M * (<*uf*>@)) by A5,MATRIX_0:23,Th2;
A13:        width M = len (<*uf*>@) by A3,MATRIX_0:23;
            reconsider a1 = 1,a2 = 0 as Element of F_Real;
A14:        Line(M,2) = <* a2,a1,a2 *> & uf =<* uf.1,uf.2,uf.3 *>
              by Th56,A1,FINSEQ_1:45;
            dom uf = Seg 3 by A1,FINSEQ_1:def 3;
            then reconsider uf1 = uf.1,uf2 = uf.2,
              uf3 = uf.3 as Element of F_Real by FINSEQ_1:1,FINSEQ_2:11;
A15:        Line(M,2) "*" uf = a2 * uf1 + a1 * uf2 + a2 * uf3 by A14,Th6
                            .= uf.2;
            1 in Seg 1 by FINSEQ_1:1;
            then 1 in Seg width (M * (<*uf*>@)) by A5,MATRIX_0:23;
            then Line(M * (<*uf*>@),2).1 = (M * (<*uf*>@))*(2,1)
                                            by MATRIX_0:def 7
                                        .= Line(M,2) "*" Col( <*uf*>@,1)
                                            by A12,A13,MATRIX_3:def 4
                                        .= Line(M,2) "*" uf by Th76;
            hence thesis by A15;
          end;
        end;
        hence thesis by A11,FINSEQ_1:40;
      end;
      thus (1.(F_Real,3) * (<*uf*>@)).3 = <* uf.3 *>
      proof
        3 in Seg 3 by FINSEQ_1:1; then
A16:    (M * (<*uf*>@)).3 = Line(M * (<*uf*>@),3) by A5,MATRIX_0:52;
        now
          thus len Line(M * (<*uf*>@),3) = width (M * (<*uf*>@))
                                             by MATRIX_0:def 7
                                        .= 1 by A5,MATRIX_0:23;
          thus Line(M * (<*uf*>@),3).1 = uf.3
          proof
A17:        [3,1] in Indices(M * (<*uf*>@)) by A5,MATRIX_0:23,Th2;
A18:        width M = len (<*uf*>@) by A3,MATRIX_0:23;
            reconsider a1 = 1,a2 = 0 as Element of F_Real;
A19:        Line(M,3) = <* a2,a2,a1 *> & uf =<* uf.1,uf.2,uf.3 *>
              by Th56,A1,FINSEQ_1:45;
            dom uf = Seg 3 by A1,FINSEQ_1:def 3;
            then reconsider uf1 = uf.1, uf2 = uf.2,
              uf3 = uf.3 as Element of F_Real by FINSEQ_1:1,FINSEQ_2:11;
A20:        Line(M,3) "*" uf = a2 * uf1 + a2 * uf2 + a1 * uf3 by A19,Th6
                            .= uf.3;
            1 in Seg 1 by FINSEQ_1:1;
            then 1 in Seg width (M * (<*uf*>@)) by A5,MATRIX_0:23;
            then Line(M * (<*uf*>@),3).1 = (M * (<*uf*>@))*(3,1)
                                            by MATRIX_0:def 7
                                        .= Line(M,3) "*" Col( <*uf*>@,1)
                                            by A17,A18,MATRIX_3:def 4
                                        .= Line(M,3) "*" uf by Th76;
            hence thesis by A20;
          end;
        end;
        hence thesis by A16,FINSEQ_1:40;
      end;
    end;
    hence thesis by A2,FINSEQ_1:45;
  end;
